Hulten's Theorem for Efficient Economies¶
The results above tell us how changes in different sectors affect aggregate output. The correct way to aggregate those changes is via Domar weights, given by the sales share of a sector in aggregate output. This result holds beyond the Domar economy we covered above, where the Cobb-Douglas benchmark makes the result exact. In general, the result holds only as a first-order approximation of the effect of a sector on aggregate output. The reason is that changes in Domar weights are second order if the allocation of the economy is already efficient. This result is known as Hulten's theorem and can be verified directly as an application of the envelope theorem. We now prove it following Carvalho and Tahbaz-Salehi (2019).
Note
Historically, Domar (1961) proposed aggregation by sales-weighting sectoral productivity changes in order to account for intermediate goods in growth accounting. Hulten (1978) showed that, under efficiency, this is exactly the right local aggregation rule. So Domar provided the index-number intuition (defining aggregation in terms of index-numbers defined as weighted geometric averages), while Hulten provided the general-equilibrium justification.
A general economy.
Consider an economy with \(S\) sectors and \(F\) primary factors, all traded competitively. Sector \(s\) produces using a constant-returns-to-scale technology,
The representative household has homothetic preferences \(u(c_{1},\dots,c_{S})\) that we make homogeneous of degree one. Let \(N_{f}\) denote the endowment of primary factor \(f\), supplied inelastically.
The planner's problem.
Since the competitive equilibrium is efficient, we can characterize it through the planner's problem
subject to the resource constraints
Let \(\eta_{s}\) be the multiplier on the resource constraint for good \(s\), and \(\xi_{f}\) be the multiplier on the factor constraint for factor \(f\). As is standard, \(\eta_{s}\) gives the marginal value to the planner of additional \(c_{s}\) (in terms of welfare),
and \(\xi_{f}\) gives the marginal value of increasing the endowment of factor \(f\).
The envelope theorem.
What is the value to the planner of an increase in the productivity of sector \(s\)? The productivity term \(z_{s}\) enters the problem only through the production function of sector \(s\). We have, by applying the envelope theorem,
It is useful to express this in terms of elasticities (multiplying by \(z_{s}/W\)),
The market economy.
The representative household solves
where \(p_{s}\) is the price of good \(s\) and \(w_{f}\) is the price of primary factor \(f\). Let \(\phi\) denote the multiplier on the household's budget constraint. The household's first-order condition is
Comparing this with the planner's first-order condition implies
Welfare and real GDP.
We start by relating welfare to expenditure in final goods. As \(u(\cdot)\) is homogeneous of degree one, Euler's theorem implies
Nominal GDP corresponds to the expenditure in final goods in the economy. Using GDP as the numeraire implies that the GDP deflator is 1, and thus we have
Hence, welfare and GDP are tied:
We can now state and prove Hulten's theorem.
Proposition 19.1: Hulten's Theorem
In an efficient economy,
where \(\lambda_{s}\,\equiv\,p_{s}y_{s}/Y\) is the Domar weight of sector \(s\) and \(\Lambda_{f}\,\equiv\,w_{f}N_{f}/Y\) is the income share of factor \(f\).
Proof
From the envelope condition for the effect of sector \(s\)'s productivity on aggregate welfare, we have
where we use the facts that \(\eta_{s}=\phi p_{s}\) and \(W=\phi Y\).
Similarly, we have for factors of production,
where we use the facts that \(\xi_{f}=\phi w_{f}\) and \(W=\phi Y\).
Therefore, the total differential of real output \(Y\) in this representative-agent economy is summarized by the homogeneous utility index, and is given by
This gives the result.
This interpretation also clarifies why gross sales are the right weights for aggregation. Value added counts a sector's contribution only for final consumption. Gross sales keep track of the fact that the same sector may matter multiple times through the network (even if it never makes it to final consumption). If one unit of a good is used in many downstream activities, its effect on aggregate output is larger than what one would infer from final-demand shares alone. This is exactly the insight behind Domar's aggregation rule, and Hulten's theorem shows that this intuition is the correct one in efficient economies.
The key insight is that Domar weights are a sufficient statistic for the technology and input-output linkages of each sector. This is surprising at first: regardless of the form of aggregate demand \(u\), the production technologies \(\{f_{s}\}\), and the input-output network \(\Omega\), the effect of a sector on aggregate output is captured entirely by its sales share. This captures how intermediate inputs affect aggregate output both directly through final demand and indirectly through the production of all downstream sectors, subsuming the information about how final goods are produced.
It turns out that the basic logic of Hulten's theorem extends beyond effects on GDP. The envelope theorem covers many other relevant cases. Baqaee and Farhi (2019) and Baqaee and Rubbo (2023) show the same result applies to real domestic absorption, welfare, or other measures of real activity.
Finally, the Cobb-Douglas environment is special. Unit elasticities keep expenditure shares fixed. Productivity shocks move prices and quantities, but they do not change the input-output coefficients or the Domar weights. Hence the same matrix \(\Psi\) summarizes propagation before and after the shock and there is no reallocation. In more general environments, shocks change the endogenous allocation matrix and therefore also the effective network itself. Then Hulten's theorem survives only as a first-order approximation. In the Cobb-Douglas case, however, the first-order formula is exact because the network weights are constant.
Example: Micro to macro fluctuations¶
Hulten's theorem can be readily applied to growth accounting or to the understanding of the variability of GDP. To see this, return to the one-factor environment and suppose that the endowment of labor is fixed, while sectoral productivity shocks
are independent and identically distributed with mean zero and standard deviation \(\sigma\). The i.i.d. assumption is strong, but helps us focus on how the input-output network amplifies micro fluctuations. If the shocks were correlated we could separate them into a common ("aggregate") component and the i.i.d. component.
Hulten's theorem implies the standard deviation of aggregate output growth due to idiosyncratic sectoral shocks is
So, the volatility of aggregate output depends on the second (uncentered) moment of Domar weights. That is, how unevenly the economy loads those shocks through the vector of Domar weights.
Now impose the additional restriction that all sectors have the same labor share, \(\alpha_{s}=\alpha\). Since labor is the only primary factor,
Hence we can rewrite the previous expression in terms of centered moments as
This expression makes more transparent the value of diversifciation. If all Domar weights are identical, then \(\mathrm{Var}(\lambda_{1},\dots,\lambda_{S})=0\) and
Thus aggregate volatility falls at the familiar \(1/\sqrt{S}\) rate. But if Domar weights are dispersed, aggregate volatility is larger. The reason is that shocks to sectors with larger Domar weights do not wash out symmetrically in the aggregate. Even when micro shocks are purely idiosyncratic, heterogeneity in sectors' sales shares can therefore generate sizable fluctuations in GDP.